# Fungrim entry: 57d31a

$\chi(n) \in \left\{ {e}^{2 \pi i k / r} : k \in \mathbb{Z} \right\} \cup \left\{0\right\}\; \text{ where } r = \varphi(q)$
Assumptions:$q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}$
TeX:
\chi(n) \in \left\{ {e}^{2 \pi i k / r} : k \in \mathbb{Z} \right\} \cup \left\{0\right\}\; \text{ where } r = \varphi(q)

q \in \mathbb{Z}_{\ge 1} \;\mathbin{\operatorname{and}}\; \chi \in G_{q} \;\mathbin{\operatorname{and}}\; n \in \mathbb{Z}
Definitions:
Fungrim symbol Notation Short description
Exp${e}^{z}$ Exponential function
Pi$\pi$ The constant pi (3.14...)
ConstI$i$ Imaginary unit
ZZ$\mathbb{Z}$ Integers
Totient$\varphi(n)$ Euler totient function
ZZGreaterEqual$\mathbb{Z}_{\ge n}$ Integers greater than or equal to n
DirichletGroup$G_{q}$ Dirichlet characters with given modulus
Source code for this entry:
Entry(ID("57d31a"),
Formula(Where(Element(chi(n), Union(Set(Exp(Div(Mul(Mul(Mul(2, Pi), ConstI), k), r)), ForElement(k, ZZ)), Set(0))), Equal(r, Totient(q)))),
Variables(q, chi, n),
Assumptions(And(Element(q, ZZGreaterEqual(1)), Element(chi, DirichletGroup(q)), Element(n, ZZ))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC